#9779764

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#37789	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26,31,34)(27,35,37,29)(28,32,38,36)(30,33,39,40);(2,5)(6,7)(11,17)(14,21)(18,23)(22,24)(25,27,31,37)(26,29,34,35)(28,33,38,40)(30,36,39,32);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,28,31,38)(26,32,34,36)(27,33,37,40)(29,30,35,39);(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,38)(39,40);(1,2)(3,7)(4,5)(6,8)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,29)(26,27)(28,30)(31,35)(32,33)(34,37)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,30)(26,33)(27,36)(28,35)(29,38)(31,39)(32,37)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9686960);